Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The zero ring has empty spectrum

Example

With the unital convention, Spec⁡(0)=∅ and the empty affine scheme has global ring 0.

Facts & Assumptions

Given: The zero ring 0, in which 0=1.

[F1]

Global functions on an affine spectrum recover its defining ring (Global functions on Spec A recover A).

Verification

technique · direct
1.1

The only ideal of the zero ring contains 1=0, hence is not proper.

given
2.1

Thus Spec⁡(0) has no points and is empty.

step 1.1
3.1

By [F1], its global sections are 0.

F1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources